no. 5
Sets, groups, and fields definable in vector spaces with a bilinear form
[Ensembles, groupes et champs définissables dans des espaces vectoriels avec une forme bilinéaire]
Dobrowolski, Jan1, 2
1 Instytut Matematyczny Uniwersytetu Wrocławskiego pl. Grunwaldzki 2/4, 50-383 Wrocław (poland)
2 School of Mathematics University of Leeds Leeds LS2 9JT (United Kingdom)
Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 1795-1841.

Nous étudions des ensembles, des groupes et des corps définissables dans la théorie des espaces vectoriels de dimension infinie sur un corps algébriquement clos de caractéristique différente de 2 munis d’une forme bilinéaire symétrique (ou alternée) non dégénérée. Tout d’abord, nous définissons une notion de dimension d’un ensemble définissable, qui possède de nombreuses propriétés de rang de Morley dans les théories fortement minimales. Ensuite, en utilisant cette notion de dimension comme outil principal, nous prouvons que tous les groupes définissables sont (algébriques-par-abéliens)-par-algébriques. Nous concluons que tout corps définissable est définissablement isomorphe au corps des scalaires de l’espace vectoriel. Nous déduisons d’autres conséquences du bon comportement de la dimension, par exemple chaque type générique dans tout ensemble définissable est un type définissable ; chaque ensemble est une base d’extension ; chaque groupe définissable a une composante connexe définissable.

Nous démontrons également des résultats analogues en travaillant sur des corps réels clos.

We study definable sets, groups, and fields in the theory of infinite-dimensional vector spaces over an algebraically closed field of any fixed characteristic different from 2 equipped with a nondegenerate symmetric (or alternating) bilinear form. First, we define a notion of dimension of a definable set, which enjoys many properties of Morley rank in strongly minimal theories. Then, using this dimension notion as the main tool, we prove that all definable groups are (algebraic-by-abelian)-by-algebraic. We conclude that every definable field is definably isomorphic to the field of scalars of the vector space. We derive some other consequences of good behaviour of the dimension, e.g. every generic type in any definable set is a definable type; every set is an extension base; every definable group has a definable connected component.

We also prove analogous results working over real closed fields.

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Révisé le :
Accepté le :
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DOI : 10.5802/aif.3559
Classification : 03C60, 03C45
Keywords: Bilinear form, definable group, definable field
Mot clés : Forme bilinéaire, groupe définissable, corps définissable
Dobrowolski, Jan 1, 2

1 Instytut Matematyczny Uniwersytetu Wrocławskiego pl. Grunwaldzki 2/4, 50-383 Wrocław (poland)
2 School of Mathematics University of Leeds Leeds LS2 9JT (United Kingdom)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Dobrowolski, Jan. Sets, groups, and fields definable in vector spaces with a bilinear form. Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 1795-1841. doi : 10.5802/aif.3559. https://aif.centre-mersenne.org/articles/10.5802/aif.3559/

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